#Evaluating Definite Integrals

#Table of Contents

1. Introduction
2. Fundamental Theorem of Calculus
3. Steps to Evaluate Definite Integrals
4. Key Concepts
5. Worked Examples
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Introduction

Evaluating a definite integral involves finding the numerical value of the integral over a specified interval. This process is crucial in calculus and has various applications in physics, engineering, and other fields.

#Fundamental Theorem of Calculus

The first fundamental theorem of calculus states that if $f$ is a continuous function on the closed interval $[a, b]$, then:

$${\int }_{a}^{b}f(x) , dx = F(b) - F(a)$$

where $F$ is an antiderivative of $f$.

#Steps to Evaluate Definite Integrals

1. Find the Antiderivative: Determine the indefinite integral of the function $f(x)$, denoted as $F(x)$. $$\int f(x) , dx = F(x) + C$$

2. Apply the Limits: Use the evaluated antiderivative to compute the difference between its values at the upper and lower limits. $${\int }_{a}^{b}f(x) , dx = {\left[F(x)\right]}_{a}^{b} = F(b) - F(a)$$

3. Ignore the Constant of Integration: When evaluating definite integrals, the constant of integration cancels out. $${\left[F(x) + C\right]}_{a}^{b} = (F(b) + C) - (F(a) + C) = F(b) - F(a)$$